Effect of Compliant Walls on Secondary Instabilities in
BoundaryLayer Transition
Ronald D. Joslinj
NASA Langley Research Center, Hampton, Virginia 23681
and
Philip J. Morris J
Pennsylvania State University, University Park, Pennsylvania 16802
Abstract
For aerodynamic and hydrodynamic vehicles, it is highly desirable to reduce drag and noise levels. A reduction
in drag leads to fuel savings. In particular for submersible vehicles, a decrease in noise levels inhibits detection.
A suggested means to obtain these reduction goals is by delaying the transition from laminar to turbulent flow
in external boundary layers. For hydrodynamic applications, a passive device which shows promise for transition
delays is the compliant coating. In previous studies with a simple mechanical model representing the compliant wall,
coatings were found that provided transition delays as predicted from the semiempirical e" method. Those studies
were concerned with the linear stage of transition where the instability of concern is referred to as the primary
instability. For the flatplate boundary layer, the TollmienSchlichting (TS) wave is the primary instability. In one
of those studies, it was shown that threedimensional (3D) primary instabilities, or oblique waves, could dominate
transition over the coatings considered. From the primary instability, the stretching and tilting of vorticity in the
shear flow leads to a secondary instability mechanism. This has been theoretical described by Herbert based on
Floquet theory. In the present study, Herbert's theory is used to predict the development of secondary instabilities
over isotropic and nonisotropic compliant walls. Since oblique waves may be dominant over compliant walls, a
secondary theory extention is made to allow for these 3D primary instabilities. The effect of variations in primary
amplitude, spanwise wavenumber, and Reynolds number on the secondary instabilities are examined. As in the
rigid wall case, over compliant walls the subharmonic mode of secondary instability dominates for lowamplitude
primary disturbances. Both isotropic and nonisotropic compliant walls lead to reduced secondary growth rates
compared to the rigid wall results. For high frequencies, the nonisotropic wall suppresses the amplification of the
secondary instabilities, while instabilities over the isotropic wall may grow with an explosive rate similar to the rigid
wall results. For the more important lower frequencies, both isotropic and nonisotropic compliant walls suppress
the amplification of secondary instabilities compared to the rigid wall results. The twofold major discovery and
demonstration of the present investigation are: (1) the use of passive devices, such as compliant walls, can lead to
significant reductions in the secondary instability growth rates and amplification; (2) suppressing the primary growth
rates and subsequent amplification enable delays in the growth of the explosive secondary instability mechanism.
1. Introduction
Research involving fiow over flexible walls was
started in the late1950's by Kramer^'^. Experimentally,
Kramer found significant drag reductions using rubber
coatings over rigid walls. Investigators in the 1960's fo
cused on the task of experimentally duplicating and the
oretically explaining Kramer's results. The majority of
these studies failed to produce any comparable results;
yet, the theoretical results laid the foundation for all fu
ture studies involving fiexible walls. Interest turned to
ward the use of compliant walls for turbulent drag reduc
tion. In the 1970's NASA^ and in the 1980's the Office of
Naval Research'* sponsored investigations involving the
use of compliant walls for the turbulent problem. Al
thought most of the results from this era were either in
conclusive or unsatisfactory, the contributions, together
with earlier results, have acted as stepping stones to the
understanding of the physically
complex fiuid/wall interaction phenomena. A compre
hensive review of the pioneering studies was given by
Bushnell, Hefner and Ash®, in particular for the tur
bulent fiow problem. More recent reviews were given
by Riley, GadelHak and Metcalfe®, GadelHak^'®, and
Carpenter^.
Motivation for the present investigation is partially
derived from the following favorable theoretical and ex
perimental results. In the early 1980's, Carpenter and
Garrad**^'** showed theoretically that Kramertype sur
faces could lead to potential delays in transition. Fur
ther, they indicated deficiencies in previous investiga
tions which may have prevented their achieving results
comparable to Kramer's. Only recently, experiments
performed by Willis*^ and Caster*^ showed favorable re
sults using compliant walls. As outlined in the above
mentioned reviews, a number of investigations in the
past ten years have been conducted involving flexible
walls. A main emphasis of these studies was to under
stand the physical mechanisms involved in the fluid/wall
interaction of transitional and turbulent flows. Most of
these studies focused on the 2D instability problem, ex
cept Yeo^'* who showed that a lower critical Reynolds
number existed for the isotropic compliant wall for 3D
instability waves. Carpenter and Morris^® and Joslin,
Morris and Carpenter^® have shown that 3D Tollmien
Schlichting waves can have greater growth rates over
compliant walls than 2D waves. However, they showed
that, even though 3D waves may be dominant, transi
tion delays are still obtainable through the use of com
pliant walls. They considered a compliant wall model
used by Grosskreutz^^ for his turbulent boundarylayer
experiments.
In this paper, the growth rates and ampliflcation
of secondary instabilities over compliant walls are pre
dicted and compared to the rigid wall results. A the
ory for secondary instabilities is used which is based on
Floquet Theory and was developed by Herbert^®. This
theory agrees remarkably well with experiments, in par
ticular those of Klebanoff, Tidstrom and Sargent^^ for
peakvalley splitting (fundamental) and Kachanov and
Levchenko^*^ for peak valley alignment (subharmonic).
The secondary instability theory is extended to allow for
3D primary instabilities, which are dominant over the
compliant walls considered. Appropriate boundary con
ditions for the Grosskreutz^^ wall model are derived for
the secondary analysis.
In the next section, the primary instability problem
is outlined. In section three, the secondary instability
problem is discussed. The numerical methods, results,
and a summary follow in the remaining sections.
2. Primary Wave Model
Results for 2D and 3D primary instabilities over
compliant and rigid walls have been presented by
Joslin^"'^, Morris and Carpenter"'^®'"'^®. Since the secondary
instability theory is based on and includes a primary in
stability analysis, brief derivations of the dynamic equa
tions and boundary conditions for the 3D primary insta
bility problem are included below. The disturbances are
represented as travelling waves which may grow or decay
as they propagate. Nonlinear coupling is ignored so that
individual components of the frequency spectrum may
be studied. Additionally, the quasiparallel assumption
is made.
Consider an incompressible laminar, boundarylayer
flow over a smooth flat wall. The NavierStokes equa
tions govern the flow. The Blasius proflle is used to
represent the mean flow.
A smallamplitude disturbance is introduced into
the laminar flow. A normal mode representation is given
as
{v[,Q[}(x, y, z, t) ={vi,Qi}(y) exp[i(a;a cos (f)
+za sin (f) — ujt)] + c.c. (1)
where vi and ili are the complex eigenfunctions of nor
mal velocity and vorticity, respectively. To obtain a real
solution, complex conjugate solutions denoted by c.c. are
required, a is the wavenumber, w is the frequency, and (f)
is the wave angle. In general, a and w are complex lead
ing to an ambiguity in the system. For temporal analy
ses, a is a real specifled wavenumber and w is the com
plex eigenvalue. For spatial analyses, w is a real specifled
frequency and a is the complex eigenvalue. For the com
pliant wall problem, Joslin, Morris and Carpenter^® have
shown that the use of eqn. (1) leads to an overestimation
of the growth of the wave as it propagates. The wave ac
tually propagates in a nearly streamwise direction which
is in the direction of the group velocity, not normal to the
wave fronts. In the present paper, the secondary insta
bilities are investigated using this simple representation
of the primary instabilities. Since the present approach
is conservative, it should exemplify the beneflts of using
compliant walls as a means to obtain transition delays.
Also, a major emphasis and motivation of the present
study is to determine the behavior, or response, of the
phenomenanamely secondary instabilitiesto compliant
walls.
If the normal mode relation (1) is substituted into
the linearized form of the NavierStokes equations, the
following nondimensional system results:
where
v'{" + ai{y)v'{ + a2{y)vi = (2)
ai(y) = — iRs{Uo{y)ci cos (j) — uj) — 2a
aziy) = iRja^ {Uo{y)a cos (f) Lu)
+iRsa cos <f)Uo"{y) + a
n'\ + a3(j/)fii + a4iy)vi = (3)
asiy) = a^  iRs(Uo(y)a cos (f)  w)
aiiy) =  i Rs a sin (t)U ' o(y)
The equations are nondimensionalized using the
freestream velocity Uoo, kinematic viscosity i>, and
an appropriate length scale. Convenient lengths for
the boundarylayer scale with the ^Reynolds number,
ijj, = Uoox/i'. These include a thickness, 6, where
the Reynolds number is deflned Rs = Rx and a
boundarylayer displacement thickness, 6* , where Rs =
where
I/O
17207i?j; . Eqns. (2) and (3) are referred to as the Orr
Sommerfeld and Squire equations, respectively. The sys
tem requires six boundary conditions. Requiring that
the disturbance fluctuations vanish at infinity supplies
three:
vi{y) , v,'{y) , fii(j/)^0
y
(4)
The remaining boundary conditions are determined from
the compliant wall model.
The compliant wall model used for the present pa
per was introduced by Grosskreutz^^ in his experimental
drag reduction studies with turbulent boundary layers.
He suggested that the link between streamwise and nor
mal surface displacements would cause a negative pro
duction of turbulence near the wall. Although his results
for the turbulent fiow were disappointing, the surface
does react to the fiuid fiuctuations in transitional fiow in
such a way as to reduce production of instability growth.
Carpenter and Morris^^ have shown by an energy analy
sis how the many competing energytransfer mechanisms
are infiuenced by the compliant wall presence. Of note
is the reduced energy production by the Reynolds stress
which may cause the reduced growth rates. Further,
Joslin, Morris and Carpenter^® predicted that transition
delays of 410 times the rigid wall transition Reynolds
number were achievable with this coating. So the model
has been extended to allow for a secondary instability
analysis.
The mechanical model consists of a thin, elastic
plate supported by hinged and sprung rigid members in
clined to the horizontal and facing upstream at an angle,
6, when in equilibrium. A sketch of the mechanical wall
model is shown in Fig. 1. The boundary conditions are
obtained by enforcing a balance of forces in the stream
wise and spanwise directions and the continuity of fiuid
and wall motion. These are given below in linearized
form.
Fig. 1 Mechanical model representing the
Grosskreutz compliant coating.
For small displacements of an element out of equi
librium, the mechanical surface can be thought to move
in a direction perpendicular to the rigid swivelarm. The
horizontal and vertical displacements (Ci,rii) are linked
to the angular displacement (66) by
^1 = £,56' sin 6* and
m
£66 cos 6
(5)
where £ is the length of the rigidarm member. Equations
of motion for the element in the streamwise and spanwise
directions may be obtained by a balance of the forces of
the fiuid fiuctuations acting on the surface and the forces
due to the wall motion. These equations are
dt^
dx^
2Bx
dx'^dz'^
+B, ^) cos2 6 + Ke m  E^h
5^6
{—P \ Tyy) COS 6 + Tyx SlU 6 COS 6
and
KsCiE,b
d\i
sin 6 cos 6
(6)
(7)
where
{Bx, Bz}
{Ex,E,}b^
12(1  V^V;,)
id Br
yBxB;,
Ci is the spanwise surface displacement, pm and h are the
plate density and thickness; (B^, B^z, B^) are the fiexu
ral rigidities of the plate in the streamwise, transverse,
and spanwise directions; {Ex, E^) are the moduli of elas
ticity of the plate; Ke,Ks are the effective streamwise
and spanwise spring stiffness; p is the pressure fiuctua
tion which is obtained from the fiuid momentum equa
tions; and Tyx, Tyy and Ty^ are the streamwise, normal,
and spanwise viscous shear stress fiuctuations in the fiuid
acting on the wall.
The terms on the left hand side of eqn. (6) refer
to mechanical forces and the terms on the right refer to
fiuid motion forces due to viscous stress and pressure
fiuctuations. For the case where the ribs are aligned
at 6 = Q° , the wall becomes isotropic and reduces to the
theoretical model studied by Carpenter and Garrad^*^'^^.
Otherwise the wall is referred to as nonisotropic and the
rib angle is determined by 6.
The continuity of fiuid/wall motion is given in the
streamwise, normal, and spanwise directions, respec
tively as
56
dt
drji
dt
dt
— — = Ml +r]iUo
Vl
Wl
(8)
(9)
(10)
where (ui,vi,wi) are the disturbance velocity compo
nents in the streamwise, normal and spanwise directions.
For the Grosskreutz coating, Ks ^ oo is assumed, which
from eqn. (7) would result in zero effective spanwise
surface displacement. This implies from eqn. (10) that
wi(0) = 0. Strictly speaking, if the assumption Ks ^ co
is relaxed, the resulting instabilities have larger growth
rates. This suggests that spanwise stiffeners are stabiliz
ing to a disturbed flow. So with the assumption enforced,
a better coating for potential transition delays results.
The surface displacement takes the same normal mode
form as the primary wave given by eqn. (1). The normal
modes are substituted into eqns. (610). The equations
can be reduced to three equations in terms of the normal
velocity and vorticity'^®'^"'^.
3. Secondary Instability Theory
In this section, a secondary instability theory devel
oped by Herbert^® is extended to allow for 3D primary
instabilities. Additionally, boundary equations describ
ing the compliant walls are introduced for secondary in
stabilities. The flow is governed by the NavierStokes
equations. Instantaneous velocity and pressure compo
nents are introduced and given as
v(x,y,z,t) = V2(x,y,z,t) + Bv3(x,y,z,t)
p{x, y, z, t) = p'zix, y, z, t) ^ Bpz{x, y, z, t) (11)
where ps and v^ = (m3,i'3,W3) are the secondary dis
turbance pressure and velocity in the fixed laboratory
reference frame (x,y,z); and p2 and V2 = (m2,^2,m'2)
are the basic pressure and velocity given by,
V2{x,y,z,t) = {Uo{y),0,0} + A{ui,vi,wi}{x,y,z,t)
P2{x,y,z,t) = Api{x,y,z,t) (12)
The basic fiow is given by the Blasius profile and eigen
functions of the primary wave. Assume locally that the
primary wave is periodic in t and periodic in {x, z) with
wavelength A^ = 2t: / a^ and efine a disturbance phase
velocity which is
Cr = {Cx = LOr I Ctr COS (/), 0, C^ = LO^ / Ctr sin (j)) .
Then in a frame moving with the primary wave,
vi{x, y, z) = vi(x, y, z) = vi(x + \x,y, z + X;,) (13)
where (x, z) is the reference frame moving with the wave.
With an appropriate normalization of primary eigen
functions {ui,vi,wi) the amplitude. A, directly mea
sures the maximum streamwise rms fiuctuation. This
is given by
max «i(j/)p = «i(j/™)P = 1/2 (14)
0<y <oo
The instantaneous velocities and pressure (11) are
substituted into the NavierStokes equations which are
linearized with respect to the secondary amplitude, B.
The disturbance pressure is eliminated, resulting in the
vorticity equations.
^V2  11 n3{v2 ■ V)fi3  (V3 ■ V)fi2
Kg Ot i
+ (^2 • V)V3 + (^3 • V)V2 = (15)
with the continuity equation
V 173 = (16)
As with the primary problem, the equations are formed
in terms of the normal velocity (^3) and vorticity {^3).
The equations are found to take the form^^,
[7^V2^(t^oC.)f + C A]fi3
^Rj ot ox OZ'
+ A{(FiV)fi3(i73V)fii
dUo dv3
dy dz
+ ( __ ^^^ __Q ^__ r _ ^^^
dz dx dy dx dz
dvi du3 dvi dw3 ~i
dz dy dx dy J
(17)
and,
iir^ 7^iUoc.)T + c,T]v'v3■^
 Rs
d_
dt
d_ d_
' dx dz"
dy"^ dx
I dy dx'^ dz'^ dy''
^d'^vi. d ,d'^vi d'^vi d'^vi „ 9^mi , d
_l_ 2 —) 1 (    \ 2 —) —
dxdy dx dz'^ dx^ dy^ dxdy dy
.d'^wi d'^wi d'^ui d'^vi . d ,dvi „9mi,
__ ( i i _ i —\ 1_ ( — 1 __ 2 — )
dy'^ dx^ dxdz dydz dz dy dx
d^ d^ d^ dvi d^ r)/9ui dwi d^
dz"^ dx"^ dy"^ dx dxdy dz dx dxdz
dvi d"^ 1 ^ r_ A^v^t; ) ^ 2(^^  ^^ F ^^
dz dydz dx dz"^ dx"^ dy"^
d'^ui d ^.d'^wi d'^vi d dvi d^ d^
dxdy dx dxdy dxdz dz dx dx'' dz''
dy'' dy dx dxdy dx dydz
dvi 9^ 1 r 9 2 \ o/'^^"! "^^^i \ ^
dz dxdz dz dydz dxdz dx
dvi , d^ 9^ , „9mi d^
dz dx'' dy'' dz dxdy
]W3}
(18)
The disturbance quantities V3, Q3 and dv3/dy are re
quired to vanish far from the wall and at the wall for the
rigid wall case. The compliant wall equations give the re
maining boundary conditions in the compliant case. Ad
ditionally, the primary amplitude, A, is a parameter in
the equations and is assumed to be locally nonvarying.
As j4 ^ the OrrSommerfeld and Squire equations re
sult. For the case of interest where A^ 0, the primary
eigenfunctions (mi, vi, wi) appear in the equations as co
efficients.
To solve the secondary problem, a normal mode so
lution having the following form is assumed,
V3{x,y,z,t) = e^'+'f^^' ""^ "^^ "''' "^Wix, y, z) (19)
where /3 = 2t: /X^, is a specified spanwise wavenumber
and (T = (Tr  iiTi is a temporal eigenvalue or is spec
ified for spatial analyses. V(x,y,z) is a function that
represents the class of secondary modes. Floquet theory
suggests the form of solution for periodic systems. For
the present problem, this may be written
V(x, y, z) = eT(^ ''°'"f'+' """^ 'f''^V(x, y, z)
(20)
where 7 = 7^ + ^Ti is the characteristic exponent and
V{x, y, z) is periodic in the {x, z) plane and may be rep
resented by a Fourier series. Thus the representation of
the secondary instability for a 3D basic fiow is.
vs =e
i7t\il3(z cos (f) — x sin (f)')\^(x cos (f)\z sin (/>)
E ''^yy^"
l2)ar{x cos (j)\z sin (/>)
(21)
n = — 00
This suggests a form of solution for the secondary distur
bance based on a coordinate system oriented at an angle
4> with respect to the mean fiow and moving with the
primary wave. If the coordinate system is aligned with
the primary wave, or (/) = 0°, then the solution for the
secondary disturbance would follow Herbert, Bertolotti
and Santos^^ who considered a 2D primary wave.
If solutions given by eqn. (21) are substituted into
eqns. (17) and (18), an infinite system of ordinary differ
ential equations result. The dynamic equations are de
termined by collecting terms in the governing equations
with like exponentials. The system consists of two dis
tinct classes of solution because the even and odd modes
decouple. Even modes correspond to the fundamental
mode of secondary instability, and the odd modes are
the subharmonic mode. Only a few terms of the Fourier
series are retained since, as shown by Herbert, Bertolotti
and Santos^^, this provides a sufficiently accurate ap
proximation for a 2D disturbance.
This form of solution indicates two complex quanti
ties, (T and 7, which leads to an ambiguity similar to
that found with the OrrSommerfeld/Squire problem.
There are four unknowns, <Tr,<^i,lr,li Two can be de
termined while two must be chosen in some other way.
In the present study, temporallygrowing tuned modes
are examined. The temporal growth rate is cr^, and <Ti
can be interpreted as a shift in frequency. In this case,
7^ = 7i = 0. If (Tj = 0, then the secondary disturbance
is travelling synchronously with the basic fiow.
The boundary conditions for the secondary distur
bance are given as
Vn,v',^n ^0
y
(22)
along with the compliant wall boundary conditions. In
the rigid wall case.
,f^n=0
at
y
(23)
The analysis for the compliant boundary conditions for
secondary instabilities follows the same route as was
taken for the primary instabilities, except a number of
additional terms arise due to the presence of the primary
wave.
The fiuid/wall motion must be continuous in each
direction. In addition, the equations of force (6, 7) must
balance in the streamwise and spanwise directions in the
reference frame moving with the primary wave. Con
sistent with the fiuid equations, the amplitude of the
primary wave is assumed to be locally nonvarying. In
deriving the final form of the wall equations, a signifi
cant difference between the primary and secondary form
arises from the pressure contribution. The pressure for
the secondary disturbance is determined from the mo
mentum equations which are complicated by primary
coupling terms.
The continuity of motion between the fiuid and solid
is given by
56
dt
 «3 + mK + ^{(6 • v)«3 + (6 • v)«i} (24)
^ = ^3 + a[{1, ■ V)vs + (6 • V)t;i } (25)
^ = ws + A[{i,V)ws + {i^V)w,] (26)
where
: „ ^ d d ^ d
ox ay oz
Equations (2426) involve six unknowns for the velocity
fiuctuation and surface displacement in a highly coupled
system. As with the primary boundary conditions, it
is possible to derive a set of equations which represent
the surface motion in terms of the normal velocity and
vorticity only. This is algebraically very tedious. A com
plete derivation is given by Joslin^"'^. Note that if A =
in the secondary wall equations, the primary wall equa
tions result. This occurred with the fiuid equations as
well.
4. Numerical Methods of Solution
The algebraic complexity of the dynamic equations
for the secondary disturbance and the compliant wall
equations requires that care be taken in applying any
numerical technique. Because no theoretical or exper
imental data are available for the compliant problem,
both shooting^® and spectraP^ approximations are used.
Also, noting that Bertolotti^'* has shown for the rigid
wall problem with a 2D primary instability that, after
the transformation from spatial to temporal, the solu
tions are in good agreement, a temporal analysis is pre
sented in this paper.
For the spectral method, Chebyshev series are intro
duced to approximate each mode of the Fourier series.
An algebraic transformation is used to change from the
Chebyshev spectral domain [—1, 1] to the physical do
main. Due to the properties of the Chebyshev polyno
mial, the equations are recast in integral form. Cheby
shev polynomials are used to represent the basic flow in
the series which are substituted into the integral equa
tions. For the basic flow, 35 polynomials provided suffi
cient resolution of the eigenfunctions. The series repre
senting the secondary instability requires 40 polynomi
als for sufficient convergence to the dominant eigenvalue.
For the shooting method, beginning with the equations
for the compliant wall, integrations of the disturbance
equations across the boundary layer are performed us
ing a RungeKutta scheme. At the edge of the bound
ary layer, the numerical solution vectors are matched
with the asymptotic solutions. A very accurate initial
guess is found to be required for convergence using this
method. To demonstrate the accuracy of the numerical
techniques, a comparison for the rigid wall case is made
with Herbert^s for Rs = 826.36, Fr = 83, /3 = 0.18,
and A = 0.02. Herbert obtained the dominant mode
a = 0.01184. In good agreement, the present spec
tral and shooting methods lead to cr = 0.011825 and
a = 0.011839, respectively.
5. Results
For all of the results that follow, the freestream ve
locity is 20 m/s, the density is 1000 kg/m^, and the kine
matic viscosity is 1 x 10~® m^/s. The coatings considered
consist of both isotropic and nonisotropic walls. Both
walls were optimized at Rs = 2240 for 2D primary
instabilities. The isotropic wall has properties 9 = 0°,
h = 0.735mm, E^ = 1.385MN/m2, K = 0.354GN/m3
and Pfn = 1000 kg/m^; and the nonisotropic wall has
properties = 60°, h = 0.111mm, E^ = 0.509MN/m2,
K = 0.059GN/m3 and pm = 1000 kg/m^. A Reynolds
number of 2240 was chosen because, for a boundary layer
over a rigid wall, the disturbance with the critical fre
quency (in the e" sense) reaches its maximum growth
rate near this value of Reynolds number. Accordingly,
this is a good choice of Reynolds number for optimizing
the wall properties. In considering 3D instabilities, the
walls optimized for 2D instabilities are used with the ad
dition of isotropic plates. The properties of an isotropic
plate are direction independent; that is, E^ = E^,. Al
though complete details of the optimization process and
philosophy are given by Carpenter and Morris^^, a recap
follows.
With a flexible wall present, other modes of instabil
ity arise. With changes in the compliant wall properties,
stable, or marginally stable, fluid and wall modes can be
come unstable and dominant. The present wall proper
ties were varied to achieve an optimal specifled condition.
This desired condition was to achieve a minimum growth
rate for a dominant 2D TollmienSchlichting instability
while keeping other modes marginally stable. For the
secondary analysis, these "optimal" compliant walls led
to no additional unstable modes. However, this is not to
say that additional growing modes may not appear for
different wall properties.
In this analysis, the primary wave amplitude {A)
and the secondary instability spanwise wavenumber (/3)
are parameters of the problem. Herbert^® showed in his
boundarylayer studies for the 2D primary wave over a
rigid wall that as the amplitude increases the growth rate
of the secondary instability increases. Also, as the span
wise wavenumber is varied, the temporal growth rate
reaches a maximum for a particular wavenumber. Ad
ditionally, Herbert showed that the subharmonic mode
reaches greater growth rates than the fundamental mode
for low amplitude disturbances. These flndings were
verifled by the direct numerical simulations of Spalart
and Yang^^. Althought both the subharmonic and fun
damental modes over the compliant walls were exam
ined, emphasis is placed on the subharmonic mode, since
as both theory and computations indicate, subharmonic
disturbances are more unstable than fundamental distur
bances for small amplitudes. Limited fundamental dis
turbance results are included to verify that these modes
do not become the dominant instability over compliant
walls.
Primary waves with frequencies that give maxi
mum disturbance growth rates are considered. For the
isotropic wall, the maximum growth rate occurs at a fre
quency Lu = 0.065 (Fr =~ 29.0), where Fr = w/Rx 10'^.
Figure 2 shows the growth rates of the subharmonic and
fundamental disturbances as a function of the spanwise
wavenumber for the rigid wall and isotropic compliant
wall. As the flgure shows, growth rates over the com
pliant wall are reduced in comparison with the rigid
wall results over the whole range of spanwise wavenum
bers. Additionally, the subharmonic disturbance has
much larger growth rates than the fundamental distur
bance, as expected. Similar trends are found in the com
parison of nonisotropic and rigid wall results. In consid
ering reductions in the growth rates of the subharmonic
mode as a result of compliant walls, the isotropic wall
suppressed the maximum growth rate by 20%. For the
maximum growth rate over the nonisotropic wall, which
occurs at a difference frequency than the isotropic wall
case, the nonisotropic wall led to a reduction of 17%
compared to rigid wall results. So for a fixed Reynolds
number and primary wave amplitude, both isotropic and
nonisotropic compliant walls lead to reduced secondary
instability growth rates compared to the rigid wall re
sults.
.010 J
f
i
.IXID
d 1 3
S^UbIH WUHBUnbd , &
Fig. 2 Growth rates of the secondary instabil
ities as a function of spanwise wavenuniber for
Rs, = 2240, Fr ~ 29.0, and A = 0.01. subharmonic:
— o— , rigid wall; • • o • •, isotropic wall and funda
mental: — X — , rigid wall; • • • x • • •, isotropic wall.
A more revealing measure of the effectiveness of us
ing compliant walls to suppress secondary instabilities is
to compute the amplitude growth and decay with down
stream distance. The amplification of the primary and
secondary instabilities are governed by
of waves at a frequency Fr = 53. At this frequency,
primary amplitudes over the the isotropic wall are sim
ilar to those over the rigid wall. This suggests that the
development of secondary instabilities might also be sim
ilar, since secondary disturbances are parametrically de
pendent on the basic fiow. For the nonisotropic wall
results shown in Figure 4, the primary amplitudes are
suppressed significantly compared to the rigid wall and
isotropic wall cases. Therefore, the nonisotropic wall
would likely lead to a very different secondary instability
development, most probably with reduced amplitudes.
Again from Figure 4, one might expect greater differ
ences in the secondary instability development over both
compliant walls as the Reynolds numbers increase and
corresponding frequencies decrease.
> 
n
r
m~
_I_L
jj
I I I I I I
_I_L
3W3
<oo sw *w
700
Fig. 3 Amplitude growth as a function of
Reynolds number for the subharmonic mode (B)
of a 2D primary wave (A) over a rigid wall at
Fr= 124, Ao = 0.0044, 5o = 1.86x 10^ and & = 0.33.
, theory and (x,o), Kachanov and Levchenko
[15].
■4
t* dx
■■4
—^dx
(27)
where Ag, Bg are the initial amplitudes aX Xg, A and B
are the amplitudes at a downstream distance x, and *
denotes dimensional quantities. As shown by Herbert^®,
the theoretical prediction of primary and secondary am
plification by eqns. (27) compares well with the exper
iments of Kachanov and Levchenko^*^. A similar com
parison is shown in Fig. 3, where the theoretical results
were obtained with the present numerical techniques.
Before computing similar amplification results over
the compliant walls, inferences of the secondary insta
bility growth may be drawn from primary instability
results. Figure 4 shows the maximum amplification of
various frequency primary waves propagating over the
rigid and compliant walls along with the amplification
J J L.
J L
0.0
^ ^^^^
hiCTmiUc munbcT, Rr ^ U*"^
1.5
Fig. 4 2D curves of maximum amplification for
TSI waves over a , rigid wall; , isotropic
wall; and , nonisotropic wall and — • — , waves
of Fr ~ 53.
To demonstrate these postulations for secondary in
stabilities, eqns. (27) are used to compute the amplifi
cation of primary and secondary instabilities over the
rigid, isotropic, and nonisotropic walls for Ag = 0.004,
Bo = lx 10^ and h = 0.15, where h = 13/ R x 10^. The
initial amplitudes are somewhat arbitrary: the present
values were selected to be close to the experiments of
Kachanov and Levchenko^*^ for rigid walls. The spanwise
wavenumber (h) was chosen near the maximum growth
rate of the secondary instability at the branches of the
neutral curve for the rigid wall case. Both primary and
secondary amplifications are shown in Fig. 5. Clearly,
the results of the secondary instability growth over the
isotropic and rigid walls are similar, as postulated, while
the nonisotropic wall significantly suppresses the sec
ondary instability growth.
9
I
ft*' 
4i.D2S
■M CI . n7 :<
lUymifali:]
Fig. 5 Amplitude growth as a function of
Reynolds number for the subharmonic mode (B)
of a 2D primary wave (A) at Fr ~ 53, Ag = 0.004,
Bo = I X 10~®, and h = 0.15 over a , rigid wall;
, isotropic wall; and , nonisotropic wall.
The growth of the secondary instability is dependent
on the parameters of the basic fiow, most probably the
primary instability amplitude (A). For example. Fig. 6
shows the amplification of the primary and secondary
instabilities over the rigid wall with properties as before
and over the nonisotropic wall with both Ag = 0.004
and Ag = 0.008. Even by doubling the initial amplitude
of the primary disturbance, the growth of the secondary
instability over the nonisotropic wall continues to be
suppressed and has not exceeded the primary amplitude
upon crossing the neutral curve. Yet, the doubled initial
amplitude (Ag = 0.008) results in a significant increase
in the secondary instability growth compared with the
lower amplitude (Ag = 0.004) results. Hence, the sup
pression of the primary instability amplitude is of utmost
importance to suppress the onset of the secondary insta
bility growth.
0,0* Q.DG CDS 0.10 0.12
14
Fig. 6 Amplitude growth as a function of
Reynolds number for the subharmonic mode (B)
of a 2D primary wave (A) at Fr ~ 53, Bg =
1 X 10~®, and b = 0.15 over a , rigid wall
with Ag = 0.004; — • — , nonisotropic wall with
Ag = 0.004; and , nonisotropic wall with
Ag = 0.008.
.MS I
.om 
.002 
.000
Ifjiimr •iTrnnmh rii P
Fig. 7 Growth rates of the subharmonic distur
bance for 2D and 3D primary waves over the
isotropic wall as a function of spanwise wavenum
ber for Rs' = 2240, Fr ~ 22.3, and A = 0.01. ,
<j, = 0«; ■■■,<j> = 10°; and , <j> = 20°.
Proceeding with investigating the effect of compli
ant walls on secondary instabilities, 3D primary waves
are introduced and are determined by the specified wave
angle ((f)). In Fig. 7, subharmonic disturbance growth
rates over the isotropic compliant wall are shown with
variation in spanwise wavenumber and primary wave an
gle ((/)). A frequency w = 0.05 (Fr ~ 22.3) is selected
since the 3D primary wave growth rate is maximized
at this frequency for the Reynolds number Rs = 2240.
Subharmonic growth rates arising from the 2D primary
wave ((f) = 0°) are clearly larger than those from the
3D waves. As the primary wave angle ((f)) increases,
the subharmonic growth rates continually become more
damped. Additionally, for oblique waves ((f) ^ 0°) the
secondary disturbances no longer travel synchronously
with the primary wave. This is shown in Fig. 8 by the fre
quency shifts that result over the isotropic wall. Similar
results occur for the nonisotropic wall. It is likely that
this shift leads to a reduced efficiency of energy transfer
from the basic flow to the secondary disturbance. This
frequency shift is as much a result of the 3D nature of
the basic flow as it is of the compliant wall influence. Re
sults similar to those of Fig. 8 were found by Balachan
dar, Streett and Malik^® for the rotatingdisk problem
which also has a 3D basic flow.
0.00 I
■Q.O? 
Q.Q*
' m _
•
X " .
\ .

\ '
\
■
\
. , 1 . . , , 1 . , , , 1
Q.O
5.1
0.2
9
e,j
Fig. 8 Frequency shift of the subharmonic distur
bance for 3D primary waves over the isotropic
wall as a function of spanwise wavenuniber for
Rs, = 2240, Fr ~ 22.3, and A = 0.01. ■■■,<j> = 10°
and , <j> = 20°.
The secondary growth rate comparison in Fig. 7 is
misleading as a result of holding the primary amplitude
(A) flxed. With a variation in wave angle ((f)), corre
sponding primary wave amplitudes result as shown by
Joslin, Morris and Carpenter^®. Then a more realistic
comparison of secondary growth rates arising from 2D
and 3D primary waves should involve amplitudes suited
to a given wave angle ((f)). One means to derive these
amplitudes is through the use of eqns. (27). In Fig. 9,
a comparison of the subharmonic growth rates over the
isotropic compliant wall is made for 2D and 3D primary
waves allowing for amplitude differences. Similar results
are found for the nonisotropic wall as well. As shown,
the 3D primary wave does lead to much larger growth
rates than a 2D wave for the compliant walls considered.
This clearly demonstrates that in spite of the loss of syn
chronization with the basic flow, 3D primary waves lead
to dominant secondary instabilities over compliant walls.
Again, the amplitude of the primary wave is of utmost
importance for determining the secondary disturbance
growth. Yet, compliant walls do lead to a reduction in
secondary growth rates compared with those for the rigid
wall. Also, recall that the amplitudes for 3D primary
waves used in these calculations were determined by a
normal mode assumption which leads to a conservative
estimation of the primary amplitude^®, and a more re
alistic lower amplitude would lead to secondary growth
rates somewhere between the 2D and 3D results shown
in Fig. 9. But, the goal here is to determine the fun
damental effect of compliant walls on secondary distur
bances.
Final amplitude calculations using eqns. (27) are
carried out for the frequency Fr ~ 53. In Fig. 10, sec
ondary amplitudes arising from the most amplifled 3
D primary wave over a nonisotropic compliant wall are
compared to the previous results from 2D primary waves
over rigid and nonisotropic walls. At this frequency, the
amplitude difference between the 2D and 3D primary
instabilites is small, yet the secondary disturbance re
sponds notably. This is an indication that small changes
in the primary instability, however slight, have a mount
ing effect on the rapidly developing secondary instability.
,0490
i.onss
.oooa
a.o
a. 2
i>.i
0,4
Fig. 9 Growth rates of the subharmonic distur
bance for 2D and 3D primary waves over the
isotropic wall as a function of spanwise wavenuni
ber for Rs' = 1760 and Fr ~ 30.2 for , <j> = 0"
with A = 0.010; • • • , ,?i = 45° with A = 0.031.
I 
5*"^ 
1
^
.^^
fr
p.na n.iH
o.oa
O.OJt o.ia O.IS
D.I4
Fig. 10 Amplitude growth as a function of
Reynolds number for the subharmonic mode (B)
of a 2D and 3D primary wave (A) at Fr ~ 53,
Bo = lx 10^ Ao = 0.004, and h = 0.15 over a ,
rigid wall — • — , 2D nonisotropic wall; and • • •,
3D nonisotropic wall.
6. Summary
In earlier studies^®'^®'^^, it was shown that 3D pri
mary instabilities theoretically dominate transition over
the compliant walls considered, yet transition delays
were found compared to the rigid wall. The present pa
per has further extended the understanding of the effect
of compliant walls on transition mechanisms in bound
ary layers. Namely, the effect compliant walls have on
secondary instabilities has been investigated. It has been
shown that the use of compliant walls can lead to reduced
growth rates and amplification of secondary instabili
ties. From both the earlier 3D primary results combined
with the present secondary findings, it should be empha
sized that the physical nature and makeup of the mech
anisms in transition are not altered by the control device
(i.e. compliant wall). Rather, only the response of that
mechanism is changed. This fact is of particular impor
tance for designing Laminar Flow Control (LFC) studies.
As an example, the behavior of a secondary instability
growth with variation in primary amplitude is well doc
umented by Herbert. As the primary amplitudes are
reduced, the excitement of the secondary instability is
delayed. Thus, active or passive devices which suppress
primary instability growth should lead to corresponding
suppression and delay of succeeding instabilities. This
has been demonstrated above with the compliant wall.
The twofold major discovery and demonstration of the
present investigation is: (1) the use of passive devices,
such as compliant walls, lead to significant reductions in
the secondary instability growth rates and amplification;
(2) suppressing the primary growth rates and subsequent
amplification enable delays in the growth of the explosive
secondary instability mechanism.
7. Acknowledgments
Support for this project was supplied by the Naval
Sea Systems Command and the Applied Research Labo
ratory Exploratory and Foundational Research Program
under NAVSEA N003988C0051 at the Department of
Aerospace Engineering, The Pennsylvania State Univer
sity. A grant for computational support was provided
by the National Science Foundation with the Pittsburgh
Supercomputing Center.
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11